End interactive session 2B
3 * x^2 + 41. Partial derivatives by hand
- Find all partial derivatives of the following:
\[f(a, z, t)=3at^2-4t+5.1az-z^3\]
Solution
\(\frac{\partial f}{\partial a} = 3t^2 + 5.1z\)
\(\frac{\partial f}{\partial z} = 5.1a - 3z^2\)
\(\frac{\partial f}{\partial t} = 6at - 4\)
- Algal biomass (\(B\), grams) in a system is modeled by the following, where \(h\) is time (hours) and \(T\) is temperature (Celsius).
\[B(h, T)=0.6Th+2.1h^2\] For water at 28 degrees, find an expression for how biomass is changing with respect to time (\(\frac{\partial B}{\partial h}\)).
Solution
\(\frac{\partial B}{\partial h}=0.6T + (2)(2.1)h\)
\(\frac{\partial B}{\partial h}=(0.6)(28) + 4.2h\)
\(\frac{\partial B}{\partial h}=16.8 + 4.2h\)
- Find the 3rd derivative of the following:
\[G(t)=2.2 + 3.1t-5.6t^4 + \frac{4.3}{t^3}\]
Solution
First derivative:
\(G'(t) = 3.1 -(4)(5.6)t^3 +(-3)(4.3)t^{-4}\)
\(G'(t) = 3.1 -22.4t^3 -12.9t^{-4}\)
Second derivative:
\(G''(t) = 3.1. - (22.4)(3)t^2 -(12.9)(-4)t^{-5}\)
\(G''(t) = 3.1. - 67.2t^2 -(12.9)(-4)t^{-5}\)
\(G''(t) = -67.2t^2 + 51.6t^{-5}\)
Third derivative:
\(G'''(t) = -(67.2)(2)t + (51.6)(-5)t^{-6}\)
\(G'''(t) = -134.4t - 258t^{-6}\)
2. Plotting a function using {ggplot2}
- Add a new Quarto doc (I’m saving mine as
day2b-exercises.qmd) to your existingeds212-day2-GHpracticerepository and follow along with the exercises, below.
Example 1:
Use keyboard shortcuts to create a function in R
In the code chunk below, we create an R function to plot the function, \(f(x) = 3x^2 + 4\). We can use a nifty keyboard shortcut to create a R function around this expression following these steps:
- Write out the expression \(3x^2 + 4\) in a code chunk:
Highlight the expression, then use the keyboard shortcut
control+option+X(Mac) orCtrl+Alt+X(Windows) to pop open a dialog box, which will prompt you to provide a name for your function (you can call iteq1).Watch your function appear! The shortcut will automatically add any variables from the body of your function (here, just
x) as arguments insidefunction():
eq1 <- function(x) {
3 * x^2 + 4
}# load packages ----
library(tidyverse)
# Define function ----
eq1 <- function(x) {
3 * x^2 + 4
}
# create data frame with range of values to evaluate function over ----
my_data_range <- data.frame(x = c(1, 100))
# Plot it as a continuous curve over a specific range using `geom_function()` ----
# `geom_function()` automatically evaluates the function over the range of x-values specified in the df and adds the resulting y-values to the plot
ggplot(data = my_data_range, aes(x = x)) +
geom_function(fun = eq1)
Example 2:
# Define function ----
eq2 <- function(x) {
2.4 - 5*x^3 + 2.1*x^2
}
# Plot it as a continuous curve over a specific range using `geom_function()` (here, we define our data frame directly inside `ggplot()` ----
ggplot(data = data.frame(x = c(-50, 50)), aes(x = x)) +
geom_function(fun = eq2)
3. Plotting derivatives
Let’s look at the function: \(C(t)=t^3\)
Let’s plot the function along with the first derivative:
# Store the function C(t) ----
ct <- function(t) {
t^3
}
# create data frame with range of values to evaluate function over ----
my_data_range <- data.frame(t = c(-5, 5))
# Plot it ----
ggplot(data = my_data_range, aes(x = t)) +
geom_function(fun = ct)
# Find the derivative ----
dc_dt <- D(expr = expression(t^3), name = "t")
dc_dt3 * t^2# Store the derivative as a function ----
dc_dt_fun <- function(t) {
3 * t ^2
}
# Then plot them together ----
ggplot(data = my_data_range, aes(x = t)) +
geom_function(fun = ct, color = "red") +
geom_function(fun = dc_dt_fun, color = "blue")
4. Partial derivatives in R
Specify the variable you want to take the derivative with respect to.
\[f(x,y,z)=2xy-3x^2z^3\]
Find all partials, \(\frac{\partial f}{\partial x}\), \(\frac{\partial f}{\partial y}\) and \(\frac{\partial f}{\partial z}\)
# Create the expression ----
f_xyz <- expression(2 * x * y - 3 * x^2 * z^3)
# Partial with respect to x ----
df_dx <- D(expr = f_xyz, name = "x")
df_dx2 * y - 3 * (2 * x) * z^3# Partial with respect to y ----
df_dy <- D(expr = f_xyz, name = "y")
df_dy2 * x# Partial with respect to z ----
df_dz <- D(expr = f_xyz, name = "z")
df_dz-(3 * x^2 * (3 * z^2))5. One more example:
Given the function: \[P(q)=2cos(3q+0.4)-5.6e^{1.4q}\]
Find the instantaneous slope at \(q = 0.8\).
# First, create the expression ----
pq <- expression(2 * cos(3 * q + 0.4) - 5.6 * exp(1.4 * q))
# Find the first derivative with respect to q ----
dp_dq <- D(expr = pq, name = "q")
# Return the first derivative ----
dp_dq-(2 * (sin(3 * q + 0.4) * 3) + 5.6 * (exp(1.4 * q) * 1.4))# Define value of q ----
q <- 0.8
# Evaluate dp_dq at that value ----
eval(dp_dq)[1] -26.038396. Stage, commit, (pull) and push your changes to GitHub
Follow the same pattern that we practiced in today’s earlier interactive session (2A), and verify that it worked by checking out your GitHub repo.